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Optimal gerrymandering with Z3

As some of you might recall from before I spent a year writing about Hypothesis, one of my other interests is voting theory. I may be doing a talk about it at some point soon, and am reading about it.

I was thinking about Gerrymandering, and I was wondering what the best (worst) possible degrees of gerrymandering were.

For example, suppose you’ve got a population of 9 split up into 3 districts of 3. If you have 6 people out of nine who want to vote blue, it’s easy to get all three seats as blue (you put 2 in each district), and it’s easy to get only two (put all 6 of your supporters into two districts), but can you be gerrymandered to get only one? It seems implausible, but how would we prove that?

Well, as you probably guessed from the title, it turns out this is one of those things where rather than proving it by hand we can just ask a computer to do it for us! (It’s actually quite easy to prove this specific case, but I was interested in more general ones).

Here’s a Z3 program to work out the answer:

Running this, the answer is that if you have 6 people then you’re guaranteed at least two seats, but if you have 5 then you can be gerrymandered down to only one seat with the opposition claiming 2 (by your opponents splitting their four across two districts so you end up with 3, 1 and 1 in each district respectively).

It takes a list of districts with their populations and a total number of people who support a party, and tries to arrange those people in a way that maximizes the number of seats that party gets.

It only supports two party systems. It wouldn’t be too difficult to expand it to a larger number of parties though. It also doesn’t work very well for large (or even medium really) numbers of districts. It handles 20 districts in about 10 seconds, but the time seems to go up extremely nonlinearly (it handles 10 in about 20 milliseconds) and it doesn’t seem inclined to ever complete with 30 districts. This might be fixable if I understood more about performance tuning Z3.

It may be possible to improve this using the Z3 Optimize stuff, but when I tried to use Optimize I segfaulted my Python interpreter so I decided to steer clear (stuff like this is why I’ve yet to seriously consider Z3 as a backend for anything interesting unfortunately). Instead this just binary searches between a set of constraints to find the best value.